Methods of stochastic neutron transport are investigated and applied to novel formulations for the neutron number distribution and the cumulative fission energy deposition distribution. We utilize two Monte Carlo algorithms: the event-based Monte Carlo (EBMC) method and the Stochastic Simulation Algorithm (SSA) to benchmark and analyze systems. We show that the SSA outperforms the EBMC for the parameter space we are interested in. We then utilize the SSA to analyze systems composed of multiple spherical regions with fast and thermal neutrons with time-dependent reactivity insertions and determine whether group-dependent number distributions approach a gamma distribution. We develop two methods for calculating the geometry-dependent transfer probabilities, the View Factor Approximation and the Sphere Point Picking Monte Carlo Method, and compare the parametric space for which the computationally superior VF approximation holds. We discuss several numerical solution methods that have been developed for solving the nonlinear adjoint transport equation satisfied by the neutron survival probability. We show the primary competitive direct solution referred to as the $\lambda$-Acceleration Method ($\lambda$AM), and introduce a new indirect solution method called the Eigenfunction Expansion Method (EFE). The space-angle shape of the solution in a nonhomogeneous planar medium equilibrate rapidly after the initiation of a chain and for practical purposes 3 modes are sufficient to accurately capture the time variation of the survival probability, with a full fission neutron multiplicity distribution, while just 1 mode gives acceptable accuracy in steady state. The order of the nonlinearity, correlated to the induced fission chain branching, has a larger effect on the solution than the number of modes retained in the expansion. In particular, the quadratic approximation, corresponding to truncation order 2 in the nonlinear fission branching terms, is accurate for near critical systems but nonlinearity orders of 4 to 5 are necessary for more strongly supercritical media. Comparison of numerical results against the $\lambda$AM establish the quantitative accuracy and computational efficiency achievable with the EFE approach.
We derive space, angle, and time-dependent single chain a source equations for the cumulative energy deposition distribution (the FPDF) in a system via the backward Master equation formulation; from which, equations of the moments are also derived. This new formulation has the benefit of not requiring knowledge of the neutron number distribution. We then compare results of the EBMC method with the direct numerical solution of the moment equations and show excellent agreement. We then show that by altering the induced fission energy deposition distribution, the first four moments are virtually the same for supercritical systems. It is shown that the FPDF itself does indeed have noticeable alterations in the high energy deposition tails of the distribution, suggesting that one may need to consider higher order moments in order to witness a noticeable difference in the respective profile. It was also shown that the multiplicity distribution model being used, where we compared the full distribution with the MCNP mean-preserving model, has an effect on the higher energy deposition region of the single chain FPDF.
Finally, we formulate the Boltzmann Master equation- a novel nonlinear adjoint transport equation satisfied by the neutron number density distribution. In a lumped system setting, we consider several numerical discretization schemes for the number distribution, which show that typical basis and test functions used in transport methods are not robust. We apply the collocation method as well as derive an analytical generalization of Bell's distribution via solution ansatz. We then expand our scope to include space and angle dependence, derived systems of equations for the aforementioned discretization schemes, and compared the results, showing excellent agreement for long enough times in supercritical systems with the Quadratic Approximation applied.
stochastic simulation algorithm, Master equations, Monte Carlo methods, Boltzmann Master equation, neutron transport
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O'Rourke, Patrick. "Modeling and Simulation of Stochastic Neutron and Cumulative Deposited Fission Energy Distributions." (2020). https://digitalrepository.unm.edu/ne_etds/94